Nonoverlapping Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems

The nonoverlapping domain decomposition methods form a class of domain decomposition methods, for which the information exchange between neighboring subdomains is limited to the variables directly associated with the interface, i.e. those common to more than one subregion. Our objective is to design algorithms in 3D for which we can find an upper bound on the condition number κ of the preconditioned linear system, which is independent of the number of subdomains and grows slowly with p. Here, p is the maximum degree of the polynomials used in the p-version finite element discretization of the continuous problem. In this paper, we survey some of the results obtained in [2]. Iterative substructuring methods for the h−version finite element, 2D p-version, and spectral elements have been previously developed and analyzed by several authors [3, 4], [6], [1],[13, 14], [11], [5], and [7, 8, 9]. However, some very real difficulties remained when the extension of these methods and their analysis to the 3D p−version finite element method were attempted, such as a lack of extension theorems for polynomials. The corresponding results are well known for Sobolev spaces, but their extension to finite element spaces is quite intricate. In our technical work, we use and further develop extension theorems for polynomials given in [1], [12], and [10] in order to prove the following bound on the condition number of our algorithm: